Thank you for a well thought out response.
While the newer word "refactoring" seems to be pretty well-defined,
I feel that the older word "rewriting" is not.
From Martin Fowler's original Refactoring book:

Refactoring is the process of changing a software system in such a way
that it does not alter the external behavior of the code yet improves
its internal structure. ...
In essence when you refactor you are improving the design
of the code after it has been written.

Refactoring is a disciplined technique for restructuring an existing body of code,
altering its internal structure without changing its external behavior.
Its heart is a series of small behavior preserving transformations.
Each transformation (called a 'refactoring') does little, but a sequence of
transformations can produce a significant restructuring.
Since each refactoring is small, it's less likely to go wrong.
The system is also kept fully working after each small refactoring,
reducing the chances that a system can get seriously broken during
the restructuring.

Hopefully, most folks will agree with those definitions.
Now it gets much harder.
For example, your opinion:

Subversion, git, and Mercurial are not rewrites of CVS.

does not agree with mine.
My personal view is that Subversion was a "rewrite" of CVS, while the other two were not.
I don't feel strongly though.
I may well be "unorthodox", as you claim, yet I was pleasantly surprised
to discover that many others, including Joel Spolsky, share my opinion.
From Joel Spolsky:

You may also want to look into Subversion, a ground-up rewrite
of CVS with many advantages.

A good example of a complete rewrite was the Subversion version control system,
whose developers started from scratch: they believed the codebase of CVS
(an older attempt at creating a version control system), was useless and
needed to be completely scrapped.

Subversion was a re-write from the grounds up done by many of the original CVS workers.
Do you think it could have been faster to replace CVS (or CVSNT) component by component,
thus yielding Subversion?

To take another example, while I view Perl 6 as a "rewrite" of Perl 5,
I suspect many monks would disagree with that view;
a couple of them have already made that plain in this thread.
Note however that Larry Wall
at least seems to view Perl 6 as a "rewrite" of Perl:

Perl 5 was my rewrite of Perl.
I want Perl 6 to be the community's rewrite of Perl and of the community.

Admittedly, that quote was taken from State of the Onion, TPC4,
and the direction of Perl 6 has changed a bit since then.
I'd be interested to know if Larry still views Perl 6 as a "rewrite" of Perl 5.

Often open source developers feel that their code requires a revamp.
This can be either because the code was written or maintained without
proper refactoring (as is often the case if the code was inherited
from a previous developer), or because a proposed enhancement or
extension of it cannot be cleanly implemented with the existing codebase.
A final reason for wishing to revamp the code is that the code "smells bad"
(to quote Martin Fowler's Refactoring book) and does not meet the
developer's standards. There are several kinds of revamps:

Refactoring implies that the code is moved from one place to another, methods, functions or classes are extracted, duplicate code is eliminated and so forth - all while maintaining an integrity of the code. Such refactoring can be done in small amounts (so-called "continuous refactoring") to justify a certain change, or one can decide on large amounts of refactoring to an existing code that last for several days or weeks.

"Partial rewrites" involve rewriting a certain part of the code from scratch, while keeping the rest of the code. Such partial rewrites have been common in the Linux kernel development, where several subsystems were rewritten or re-implemented from scratch, while keeping the rest of the code intact.

Complete rewrites involve starting the project from scratch, while possibly still making use of some old code. A good example of a complete rewrite was the Subversion version control system, whose developers started from scratch: they believed the codebase of CVS (an older attempt at creating a version control system), was useless and needed to be completely scrapped. Another good example of such a rewrite was the Apache web server, which was almost completely re-written between version 1.3.x and version 2.0.x.

Apart from arguing over semantics, the interesting strategic decision
we face is whether to extend an existing legacy code base
or throw it away and start from scratch.
There is no one "right" answer to that question: it depends on the
project, the team, the quality of the existing code base, and many other factors.
Perhaps the most important thing is striving to prevent legacy
code degenerating into a tangled mess in the first place.

To take another example, while I view Perl 6 as a "rewrite" of Perl 5, I suspect many monks would disagree with that view; a couple of them have already made that plain in this thread. Note however that Larry Wall at least seems to view Perl 6 as a "rewrite" of Perl:

Perl 5 was my rewrite of Perl. I want Perl 6 to be the community's rewrite of Perl and of the community.

Sorry to be pedantic--it's not usually my thing--but I think you subtly reinterpreting Mr Wall's words in support of your argument.

The man himself will set me straight if it is of interest to him, but I think that "Perl 6 to be the ... rewrite of Perl" is considerably different from "Perl 6 as a "rewrite" of Perl 5".

'Perl', unadorned by the version number, is neither an implementation that can be re-written, nor a design evolution that can be reimplemented. It is a 'only'--and precisely completely--a concept; an ethos; an idea.

As such, Perl 5 wasn't a rewrite of the Perl 4 implementation; but rather a rewrite of the Perl design that was then implemented as Perl 5. Ditto for Perl 6 relative to Perl 5.

The (one; but a good one) definition of 'rewrite' in the context of software is:

A rewrite in computer programming is the act or result of re-implementing a large portion of existing functionality without re-use of its source code. When the rewrite is not using existing code at all, it is common to speak of a rewrite from scratch. ..

On the basis of both that definition, and my limited expereince of both, calling the feature rich Subversion a rewrite of the CVS, is like calling the Ford Focus a rewrite of the Ford Model-T. They serve a similar niche and target audience; but the way they go about achieving it is so utterly different.

The goal of re-implementing the same basic functionality is present; but the provision of so much additional functionality makes the term 'rewrite' an inadequate description of the reality.

Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.

"Science is about questioning the status quo. Questioning authority".

In the absence of evidence, opinion is indistinguishable from prejudice.

No. Despite my location, I'm not a betting man. But if I was, I'd confine my flutters to things with better odds of predictability that an American's use of English--pachinko, spread bets or snow on christmas day :)

More seriously. To my mind a re-write implies writing, something that was already written, again.

Perl 6 had (has?) never been written before, so it cannot be a rewrite.

Examine what is said, not who speaks -- Silence betokens consent -- Love the truth but pardon error.

"Science is about questioning the status quo. Questioning authority".

In the absence of evidence, opinion is indistinguishable from prejudice.

My intent was not to start an argument over semantics nor over anything else. I merely intended to clarify where I think some imprecision and unnecessary disagreement has entered the thread. If we keep using words we define differently as a basis, then we at least need to know how those words are being used by each party. Otherwise we'll talk past one another and nobody really knows where we would agree and disagree no matter how civil or friendly the discussion.

I also think it helps to remember that intentions toward a project can change over time. What one thinks will be a straightforward rewrite from the beginning can change in focus and gain features before the rewrite is done (or even really started). The new design can be a totally different sort of beast from the old, but since it's still in the same lineage the distinction is blurred. In fact, I suspect that the svn folks intended to rewrite CVS but looking back would only loosely use that term for what they finally did. I think Larry would say Perl 5 is a rewrite of Perl 4 from the point of view of both the language and the perl tool. I would probably say that, anyway. I think he intended originally for Perl 6 to be a rewrite of some sort, but the language is the only thing being rewritten IMO. I think Rakudo and Parrot are definitely not rewrites of perl 5.6 or 5.8 although the language implemented is still in the Perl family. How Larry actually does view things of course would be for Larry to say no matter what I think he might say.

What one thinks will be a straightforward rewrite from the beginning
can change in focus and gain features before the rewrite is done.

Yes, I suspect this happens rather a lot.
I once inherited an "unmaintainable" build system written as a huge DOS .BAT script.
Well, it was unmaintainable to me, because I didn't know .BAT very well and,
frankly, didn't want to. My strong opinion, expressed in Unix shell versus Perl,
is that you should not write non-trivial systems in .BAT (or Unix shell).
Luckily, it was only a few thousand lines long and I was able to "rewrite"
it in Perl fairly painlessly.
Because the original design and interface was so bad, I "improved" it as I went,
so my "rewrite" ended up a fair bit different to the original.
Naturally, I didn't write it as a monolithic script, but as a number of modules
along with a small script mainline.
Now you and BrowserUk may claim I didn't rewrite it, I wrote a new build system.
Fair enough, but I prefer not to argue about that any more. :)
The important strategic question is:
should I have "rewritten" it in Perl or improved it by changing
the existing .BAT script?
From my (biased) point of view, the "rewrite" was a raging success because we
were able to extend this system many times over the years (often by adding
new modules) and I feel the cost of the rewrite was got back many times
over by improved robustness and performance, along with much easier maintenance
over a period of many years.

Of course, rewriting small systems is easy.
Suppose this system has now grown to 100,000+ lines of Perl and the
person who takes it over dislikes Perl, claims it is a tangled mess,
and decides it would be more "maintainable" to rewrite it in Ruby or Python.
Is that a wise decision?
Though I would normally argue against that,
others have been known to argue for it (BTW, as far as I'm aware, sanity prevailed and Bugzilla is still written in Perl).

Whether it makes sense to rewrite depends on many factors:
how large the system is; how ambitiously you want to extend it; how clean is its code; and how
skillful are the rewriters.
I make the last point because unfortunately I've seen a number of
over-confident programmers over the years complain loudly
about a crappy old system, boldy rewrite it ... then end up with
a crappier system than the original!
In my experience, the quality of the developers doing the rewrite
is crucial:
a creaky old legacy system written by first-rate developers is
likely to be better than a shiny new one rewritten by mediocre developers.
As
Joel Spolsky warns when you rewrite:
"there is absolutely no reason to believe that you are going to do a better job".